Structural Engineering & Statics
where the load goes
Dynamics

A wall that is allowed to lift

A block that rocks inherits everything that decides whether it survives — a restoring moment set by its weight and shape that falls as it leans, and a loss of energy set by its proportions at each landing. Put a tendon through a wall and yielding bars across its base and both become design quantities, and the ratio between them decides whether the wall comes home.

Assumes The block that is safer for being bigger and The load put on backwards.

16 min read 8 figures The load that will not hold stillThe statics of things that do not moveWhich failure arrives first

A block that rocks on its base survives an earthquake by moving as a rigid body rather than by damaging itself, and the only energy it ever loses is lost when it lands on its other corner. Both of those are inherited. The restoring moment comes from the block’s weight and shape, and the loss at each landing comes from its proportions. Nothing about either was chosen, and the second is a number no test on a real base can supply in advance.

That essay ended with what a designer does about it: provide re-centring from something other than gravity, usually a tendon, and dissipation from something other than the landing, usually a yielding bar. This essay is those two sentences worked through, because each one changes the wall’s behaviour in a way that is not obvious from the sentence.

The same pulse on a wall that is designed to rock. Rotation as a fraction of the toppling angle under one 0.8 s sine pulse of 1.00 g, for the same 2.0 × 8.0 m wall of 400 kN three ways. Bare, it lifts at 0.250 g; with a 600 kN tendon it lifts at 0.625 g. The bare wall reaches 79 per cent of its toppling angle with one landing; with the tendon it reaches 27 per cent of its toppling angle with eight landings; with tendon and bars it reaches 12 per cent of its toppling angle with 43 landings. With its bars it comes to rest upright. The landings are where the bare wall loses energy; the bars add a loss that does not wait for a landing.
Fig. 1 A wall 2 m wide and 8 m tall weighing 400 kN, under one 0.8 s pulse of 1.0 g, three ways. Bare, it lifts at 0.25 g and reaches 79 per cent of its toppling angle. With a 600 kN tendon through its centre it lifts at 0.625 g and reaches 27 per cent. With yielding bars across its base as well it reaches 12 per cent, landing 43 times, and comes to rest upright.

What the wall inherits

Take the wall bare first, as the block it is. It stands on its base until the ground’s acceleration reaches b/h, a quarter of gravity for these proportions, and then rotates about a corner.

A restoring moment that gets smaller the further it leans. Restoring moment against rotation for a block 2.00 m wide and 8.00 m tall, both normalised — the moment by its value at first uplift, the rotation by the angle α = 14.0° at which the block topples. It is mgR·sin(α − θ), which is a weight times a geometry with no material property in it at all, and it has two features an elastic system does not. There is a jump at the origin: the moment is whatever the ground demands until uplift and then it is mgR sinα, so the law is discontinuous where a spring's is steepest. And the slope is negative — the further it leans the less it pushes back — so the equilibrium at θ = 0 is stable only because of that jump, and there is no stiffness to divide into a mass. The straight line is what a spring of the same first-uplift strength would have done.
Fig. 2 The restoring moment of the bare wall against its rotation, both normalised. It jumps to its full value at uplift and falls from there, reaching zero at the toppling angle of 14.0°: a weight times a geometry, with nothing in it that pushes back harder as the wall leans further.

The moment that holds the wall up is its weight times the horizontal distance from the pivot corner to its centre of mass, and that distance shrinks as the wall rotates. The further it leans, the less it pushes back. In a spring the opposite is true, and every intuition about oscillation that comes from springs is wrong for this object. There is no period, only a period for a given amplitude, and nothing in the restoring law grows to meet a larger disturbance.

So a bare rocking wall is stabilised by exactly two things: the jump at uplift, which a disturbance has to exceed before anything moves, and the energy taken out at each landing, which is all that stops a rotation once it has started. Neither can be adjusted without changing the wall’s shape.

A tendon makes the moment rise

Put a tendon down the centre of the wall, anchored at its top and in the foundation, unbonded so that it can stretch over its whole length, and prestress it to 600 kN before the wall carries anything.

At rest the tendon simply adds to the weight. It is a load put on backwards, pressing the wall onto its base with a force no gravity supplies, and the wall now lifts only when the ground’s moment beats 400 plus 600 kN times the half width, which is at 0.625 g rather than 0.25. That is the first effect and it is the smaller one.

It is nonetheless a choice, and the first one the designer makes. The same wall with a 100 kN tendon lifts at 0.313 g; with 600 kN at 0.625 g. The lift-off acceleration is (W + P₀)b/(Wh), and of the four quantities in it only P₀ is free once the wall’s size is fixed. Too little, and the wall opens under the everyday lateral loads it ought to carry closed. Too much, and it behaves as a fixed-base wall until the earthquake is large enough to damage its toe, which is the damage rocking was chosen to avoid.

The second is what happens after uplift. When the wall rotates about its toe, the base opens under the tendon by the half width times the rotation, the tendon stretches by that much, and its force rises with its stiffness. Its moment about the toe, the force times the half width, therefore grows as the wall leans.

A tendon turns a moment that falls into one that rises. Restoring moment against rotation for a wall 2.0 m wide and 8.0 m tall weighing 400 kN, three ways. As a bare block it lifts at 400 kN·m and the moment falls as it leans — 376 kN·m at 15.0 mrad. With a central tendon prestressed to 600 kN it lifts at 1000 kN·m, and because the tendon stretches as the base opens the moment rises instead, to 1276 kN·m. With yielding bars of 150 kN added at 0.6 m either side of the centre it reaches 1576 kN·m. The first line is inherited from the wall's weight and shape; the other two are chosen.
Fig. 3 Restoring moment against rotation for the wall three ways. Bare, it lifts at 400 kN·m and falls to 376 kN·m by 15 mrad. With the tendon it lifts at 1000 kN·m and rises to 1276 kN·m, because the tendon stretches as the base opens. With yielding bars as well it reaches 1576 kN·m.

That single change turns a restoring law with a negative slope into one with a positive slope, and with it the wall acquires everything a negative slope denied it: a larger disturbance now meets a larger resistance, and a rotation that starts has something that increasingly pulls it back. The weight’s contribution still falls. At these proportions it is swamped by a tendon whose stiffness adds 20 kN for every millimetre the base opens.

The tendon is also the component that makes the wall come home. Whatever else happens at the base, a stretched tendon pulls the wall toward upright with a force at least as large as its prestress, and it does so without yielding and without dissipating anything — which is exactly what it must not do, since a tendon that yielded would lose its prestress permanently and with it the re-centring.

Bars take the energy between landings

The tendon stores energy and gives it back. Something else has to remove it, and the bars are that something: short lengths of mild steel crossing the base at 0.6 m either side of the centre, anchored into the wall above and the foundation below, designed to yield.

As the base opens, the bar on the lifting side stretches by 1.6 m times the rotation and the bar nearer the toe by 0.4 m times it. At 200 kN for every millimetre of stretch, each reaches its 150 kN yield force at 0.75 mm: the far bar at half a milliradian, the near bar at 1.9. From then on each stretches plastically. When the wall rotates back, the bars are carried back through their elastic range into compression and yield again the other way — yielding one way and then the other, the cycle a buckling-restrained brace is built for.

The loop the bars draw, and the moment left when the wall closes. Moment against rotation for the wall carried through one full cycle to ±15.0 mrad and back. The moment jumps from +1000 kN·m to −1000 kN·m as the wall passes upright, because the pivot changes corner, and between those jumps the bars yield on the way out and yield back on the way in, which is what gives the loop its width. The loop encloses 16.5 kN·m, exactly the energy the bars dissipate, an equivalent damping ratio of 11.1 per cent. When the wall first closes after its excursion, 700 kN·m is still pushing it upright, so it comes back to upright with nothing holding it.
Fig. 4 Moment against rotation through one full cycle to ±15 mrad. The moment jumps between +1000 and −1000 kN·m as the wall passes upright and the pivot changes corner; between the jumps the bars yield out and yield back, which gives the loop its width. It encloses 16.5 kN·m, an equivalent damping ratio of 11.1 per cent, and 700 kN·m is still pushing the wall upright when it closes.

The loop has the shape that gives these systems their name: a flag, with a vertical staff at upright and a pennant on each side. Its area is exactly the energy the bars dissipate, because the weight and the tendon give back everything they store around a closed cycle. And it is an energy removed continuously as the wall moves, not at an instant, which is the difference that matters beside a landing. Damping that depends on landings damps slowly at exactly the large amplitudes where a wall most needs it, because large rotations mean long half-cycles and few landings. The bars do the most work when the rotation is largest.

In the pulse above, the bars are why the wall with both devices reaches 12 per cent of its toppling angle where the tendon alone reached 27. They are also why it lands 43 times: the rotations are small and quick, and the wall passes upright often.

The landing is still there, and no longer carries the design

The landings have not gone. Every wall, bare or not, loses the same share of its motion as it passes upright: the classical value for these proportions multiplies the angular velocity by 0.912, which keeps 83 per cent of the kinetic energy and takes out 17. What changed is what the landing is measured against.

A landing removes energy only from motion, and only at an instant. The wall crosses upright fast after a large excursion and slowly after a small one, and at every crossing the landing takes its 17 per cent of whatever kinetic energy happens to be there. The bars take energy out of displacement, all along the excursion, whether the wall is moving quickly or not. One depends on the proportions nobody chose; the other on a yield force somebody specified.

The loop’s 11.1 per cent is the ordinary way of putting a number on that: the energy dissipated in one cycle over 4π times the strain energy at the peak, the same measure a structure’s damping is read from when its loop is an ellipse. Here the strain energy is half of 1576 kN·m times 15 mrad, which is 11.8 kN·m, and 16.5 over 4π times 11.8 is 0.111. It is an equivalent ratio rather than a viscous one, describing a loop with a sharp staff at upright rather than an ellipse, but it is the number that lets a rocking wall be compared with a conventional wall that damps by cracking and yielding.

The ratio that decides whether it comes home

A tendon pulls the wall upright; yielded bars resist its return. Whether it gets there is a comparison between them, and it has a conventional form.

When the wall closes, weight and tendon push it home with a moment of (W + P₀) times the half width. If both bars have yielded back in compression they resist with their yield force times their two lever arms, which add up to twice the half width. The wall re-centres when the first is at least the second:

λ=W+P02Fd  ≥  1\lambda = \frac{W + P_0}{2F_d} \;\ge\; 1

For the wall above that ratio is 3.33, comfortably above one, and the closing moment of 700 kN·m in the loop is the margin. Strengthen the bars or weaken the tendon and the margin shrinks.

The loop the bars draw, and the moment left when the wall closes. Moment against rotation for the wall carried through one full cycle to ±15.0 mrad and back. The moment jumps from +500 kN·m to −500 kN·m as the wall passes upright, because the pivot changes corner, and between those jumps the bars yield on the way out and yield back on the way in, which is what gives the loop its width. The loop encloses 37.6 kN·m, exactly the energy the bars dissipate, an equivalent damping ratio of 25.3 per cent. When the wall first closes after its excursion, the moment at upright is 1 kN·m pushing it away from upright: unloaded slowly from this excursion with nothing holding it, it stops leaning 9.2 mrad.
Fig. 5 A weakly re-centred version: a 100 kN tendon against 400 kN bars, a ratio of 0.625. The loop is fatter — 37.6 kN·m, an equivalent damping ratio of 25.3 per cent — but at the moment the wall closes the moment at upright is pushing it away from upright. Unloaded slowly from this excursion with nothing holding it, it stops 9.2 mrad off upright.

That is the design trade in one figure, and it is the whole reason the ratio exists. Stronger bars take more energy out and fight harder against the wall coming home. The fatter loop is bought with the re-centring, and a wall designed for maximum damping is a wall designed to be left leaning.

What makes the comparison exact rather than suggestive is that nothing else changed. At 15 mrad both walls carry the same peak moment of 1576 kN·m. The weak wall gave up 500 kN of prestress, which is 500 kN·m about the toe, and took it back as 250 kN more in each bar, which is 250 kN times the two lever arms of 1.6 and 0.4 m — the same 500 kN·m. Same strength, same peak, more than twice the damping, and the difference between them is only where the moment comes from: a tendon that gives back what it stores, or a bar that does not.

The ratio of one is a statement about large rotations

The derivation of λ assumed something that is not always true: that both bars have yielded back in compression by the time the wall closes. The bar on the lifting side stretches four times as far as the bar near the toe, and after a small rotation the near bar may never have stretched far enough to yield back at all.

The re-centring ratio of one is a statement about large rotations. The re-centring ratio (W + P0)/2Fd above which the wall always comes back upright after one excursion, against the rotation that excursion reached, found by trying every ratio from 0.2 to 1.6 at each rotation. The conventional rule is a ratio of one, drawn dashed, and it assumes both bars have yielded back in compression by the time the wall closes. At 15.0 mrad, 20.0 mrad, 30.0 mrad they have, and the threshold is one. It is 0.80 at 5.0 mrad, 0.80 at 7.5 mrad, 0.88 at 10.0 mrad: at those rotations the bar nearer the toe never stretched far enough to yield back, so it resists closing with less than its yield force. At 3.0 mrad no ratio tried leaves the wall leaning at all, drawn on the axis. And at 5.0 mrad, 7.5 mrad the very strongest bars re-centre it too, below the leaning range, because bars that strong never yield at that rotation and give back everything they store.
Fig. 6 The ratio above which the wall always comes back upright, against the rotation it reached, found by trying every ratio from 0.2 to 1.6. After 15, 20 and 30 mrad it is exactly one. After 5 and 7.5 mrad it is 0.80, and after 10 mrad 0.88, because the bar near the toe never yielded back. After 3 mrad no ratio tried leaves the wall leaning at all.

The conventional rule is exact for a wall that has rocked far, and conservative for one that has not. A bar that has not yielded back resists closing with less than its yield force, so a lower ratio will do. At the smallest rotation drawn neither bar has yielded back, and the wall comes home whatever the ratio.

The same chart holds a second surprise at 5 and 7.5 mrad. There the wall leans only over a middle range of ratios, and the very strongest bars re-centre it again, because bars that strong never yield at that rotation at all: they stay elastic, and an elastic bar gives back everything it stores. The re-centring problem is therefore not simply “too much bar”. It is too much bar that has yielded, which depends on how far the wall went.

What a pulse does that a slow unloading does not

Every number in the last two sections comes from pushing the wall out once and letting it back slowly, which is how λ is derived and how a designer checks it. An earthquake does not unload a wall slowly, and it does not push only one way.

The same pulse on a wall that is designed to rock. Rotation as a fraction of the toppling angle under one 1.0 s sine pulse of 1.50 g, for the same 2.0 × 8.0 m wall of 400 kN three ways. Bare, it lifts at 0.250 g; with a 600 kN tendon it lifts at 0.625 g. The bare wall overturns; with the tendon it reaches 77 per cent of its toppling angle with eight landings; with tendon and bars it reaches 60 per cent of its toppling angle with 22 landings. With its bars it comes to rest upright. The landings are where the bare wall loses energy; the bars add a loss that does not wait for a landing.
Fig. 7 The same wall under a stronger pulse, 1.0 s of 1.5 g. Bare, it overturns. With the tendon it reaches 77 per cent of its toppling angle; with tendon and bars 60 per cent, landing 22 times, and it comes to rest upright. The design that survives a pulse the inherited wall cannot is the one whose restoring moment rises.

Under a pulse the weakly re-centred wall does something the slow unloading cannot show.

The same pulse on a wall that is designed to rock. Rotation as a fraction of the toppling angle under one 1.0 s sine pulse of 1.50 g, for the same 2.0 × 8.0 m wall of 400 kN three ways. Bare, it lifts at 0.250 g; with a 100 kN tendon it lifts at 0.313 g. The bare wall overturns; with the tendon it reaches 95 per cent of its toppling angle with seven landings; with tendon and bars it reaches 52 per cent of its toppling angle with nineteen landings. With its bars it cannot rest exactly upright: the bars are left in compression and it comes to rest 0.5 mrad off it. The landings are where the bare wall loses energy; the bars add a loss that does not wait for a landing.
Fig. 8 The weakly re-centred wall under the same 1.5 g pulse. With its 100 kN tendon alone it reaches 95 per cent of its toppling angle; with the strong bars 52 per cent. It does not come to rest upright, and it does not come to rest 9.2 mrad off: its bars are left in compression, and it comes to rest 0.5 mrad off upright.

The wall rocks both ways. Each excursion stretches the bar on the lifting side, and each return pushes it back; after several of them in both directions both bars carry nearly the same plastic stretch, and nearly the same compression when the wall is near upright. That symmetry brings the wall back close to upright — the rotation it cycled through undid most of the lean a single excursion would have left.

It cannot bring it exactly there. With both bars in compression at upright, they push the wall away from upright on whichever side it is, and when that push beats the weight and the tendon, upright is not a place the wall can rest. It settles instead at a lean either side, where the push and the pull balance: here half a milliradian. The bars are carrying a force no load is applying, locked in by the sequence of rotations the wall went through — the same kind of force a frictional support is left carrying by the order its loads arrived in. A residual state after an earthquake is a record of that earthquake, not a property of the wall.

It has to rock before it slides

Everything above assumes the base holds the wall sideways while it lifts. A wall that slides on its base does not rock, and which of the two a body does is decided by comparing the friction available with the ratio of its width to its height.

The tendon might be expected to change that comparison, since it adds 600 kN of clamping to the base. It does not, at least not at the moment that matters. At lift-off the horizontal force on the wall is its mass times 0.625 g, which is 250 kN, and the force clamping the base is the weight plus the prestress, 1000 kN. The ratio is 0.25. For the bare wall at its own lift-off it is 100 kN against 400 kN, also 0.25. The tendon raises the shear the wall must carry before it lifts exactly as much as it raises the clamping that resists it, so the friction a wall needs in order to rock rather than slide is b/h whatever it is prestressed to.

After lift-off the ratio barely moves. Pushed statically to 15 mrad, the moment of 1576 kN·m needs a horizontal force of about 393 kN at mid-height, and the clamping has risen to 1600 kN because both the tendon and the bars are now in tension across the base. The ratio is still a quarter. Design codes use friction coefficients of about 0.6 for concrete cast against hardened concrete, so a wall of these proportions has more than twice the friction it needs, and a squatter wall has less margin. Where it runs short, the base carries a shear key, which must hold the wall sideways without holding it down.

The price of each part, stated plainly

A designed rocking wall is a set of trades, and each device has a price that the figures above make visible.

The tendon buys a rising moment and costs re-centring margin if it relaxes. Its prestress falls with time through relaxation of the steel and creep of the concrete under the prestress, which shortens the wall and slackens the tendon, and a strand’s stress leaks away under exactly the kind of sustained stretch this one carries. A tendon that loses a quarter of its force loses a quarter of the moment pushing the wall home.

The bars buy damping and cost the return. Every kilonewton of yield force added to the bars adds area to the loop and subtracts from the closing moment, and the ratio of one marks where the second overtakes the first after a large rotation.

The bars also buy a finite life. A bar that yields in tension and then in compression on every cycle is in the regime that never settles into an elastic state: alternating plasticity, which spends a share of the bar’s low-cycle fatigue life at every excursion. That is the reason they are short, accessible and bolted rather than cast in. The wall is designed to be undamaged by the earthquake; the bars are designed to be the only thing that is.

And the rocking itself buys a displacement without damage. An earthquake asks a structure for a displacement, and a wall that supplies it by opening a gap at its base supplies it without the plastic hinge a conventional wall would form. The bars are the only parts that yield, and they are designed to be replaced.

What the idealised wall leaves out

The wall and the base are rigid. A real toe crushes as it carries the whole weight and tendon force on a small area, and a crushed toe shortens the lever arm of every restoring moment above. The landing is not Housner’s rigid impact either, which is the number nobody measures in a new place.

The tendon stays elastic. If a large rotation stretches it past yield, it loses prestress permanently, and the wall’s re-centring after that earthquake is less than it was before it.

The bars neither harden nor buckle. Real mild-steel bars harden as they cycle, which raises their force in compression when the wall closes and erodes the margin λ was computed on; and a bar pushed back into compression buckles unless it is restrained along its length, which is what a buckling-restrained brace’s casing is for.

The motion is planar and the pulse is single. A real wall rocks about two axes under a record with many pulses of varying sign, and its state when the next one arrives is the starting point for it.

Nothing damps the elastic range. Between the bars’ yields and the landings the model has no loss at all, which is why the weakly re-centred wall rests at a definite lean instead of settling slowly toward it; any real damping would bring it to the same place more gently.

Still open: how much of the tendon’s prestress survives the rocking it is there to undo

Every re-centring number above assumes the tendon’s 600 kN is still there. A tendon in a rocking wall is stretched further every time the wall rocks, and the largest stretches arrive in the earthquake that most needs the wall to come home. If the design keeps the tendon elastic at the largest rotation, the prestress is only eroded slowly, by relaxation and by the wall’s own shortening. If an earthquake larger than the design one stretches it past its limit, the wall comes out of that earthquake with less prestress than it went in with, and every λ computed for the next one is optimistic. How much margin a tendon needs to guarantee that a wall designed to re-centre still re-centres after the earthquake that tested it — and whether that margin is a stress limit on the strand or a limit on the rotation the wall is allowed to reach — is a question about the tendon’s history, not its prestress.

Where this goes next

One pulse, two blocks of the same shape. Rotation as a fraction of the toppling angle, under a single 0.8 s sine pulse of 1.00 g, for two blocks of identical proportion whose sizes differ by a factor of 3. Both lift off at the same instant, because uplift depends on the shape alone. The small one reaches the toppling angle and goes over; the large one reaches 46%. The kinks are impacts: there is no dashpot anywhere in this model, and the only energy the block loses is lost when it lands on its other corner, at a velocity ratio of 0.926 per landing — 14% of the energy each time, decided by the block's shape and by nothing else. The only damping is the landing A rocking block has no dashpot in it. The only energy it ever loses is lost at the instant it lands on its other corner, and how much is a property of the block's proportions — 14 per cent for a slender one and 38 for a stocky one. That single number decides whether it settles or goes over, and a real base does not deliver the value the theory computes. A restoring moment that gets smaller the further it leans. Restoring moment against rotation for a block 0.90 m wide and 4.20 m tall, both normalised — the moment by its value at first uplift, the rotation by the angle α = 12.1° at which the block topples. It is mgR·sin(α − θ), which is a weight times a geometry with no material property in it at all, and it has two features an elastic system does not. There is a jump at the origin: the moment is whatever the ground demands until uplift and then it is mgR sinα, so the law is discontinuous where a spring's is steepest. And the slope is negative — the further it leans the less it pushes back — so the equilibrium at θ = 0 is stable only because of that jump, and there is no stiffness to divide into a mass. The straight line is what a spring of the same first-uplift strength would have done. The block that is safer for being bigger A block resting on the ground lifts off at an acceleration that depends only on its shape and not at all on its size, and then falls over at one that depends strongly on its size. Two objects of identical proportion begin rocking at the same instant and only the smaller one topples — which is why the slender water towers stood in Chile and the squat tanks beside them did not. The loop a brace has when it cannot buckle. Force against axial deformation for two braces with the same core area, cycled six times at a storey drift of 2 per cent. An ordinary brace yields at 900 kN in tension and buckles at 482 in compression — 54 per cent of it — and the buckled shape leaves a plastic hinge that does not straighten, so the compression side loses capacity every cycle and is at 12 per cent of its first value by the last. A restrained brace has a casing that carries no axial force at all and only holds the core straight, which decouples axial capacity from flexural stiffness — the coupling that makes a strut weaker than a tie — so it yields at the same force both ways and hardens instead. The energy dissipated is 2.07 times as much over the six cycles, and the casing has to satisfy one inequality: π²EI/L² above the fully hardened core force, 2.56 here, which is a buckling check on a member carrying nothing. The brace that yields both ways An ordinary diagonal yields in tension at its full strength and buckles in compression at half of it, and the buckle leaves a hinge that does not straighten. Stop it buckling with a sleeve that carries no load at all and the loop becomes symmetric. Same deck, same load, and two pier forces. A deck bearing on a pier of 20 kN/mm with μ = 0.03, taken to the same final state two ways: the deck moves 4 mm over the pier, and the bearing's load rises from 2000 to 4000 kN. Moved first, while the bearing carries 2000 kN, the pier force reaches the limit of 60 kN and the bearing slides for the rest of the movement; the load arriving afterwards raises the limit and changes nothing, and the pier is left carrying 60 kN. Loaded first, the limit is 120 kN before the deck moves, the bearing grips throughout, and the pier carries 80 kN. Both states are at the same displacement under the same load, and both satisfy equilibrium and the friction bound; the order is the only difference, and it appears in neither. The order the loads arrived in Statics allows a contact with friction a whole range of forces and has no way to choose between them. A real structure does choose, and what it chooses by is the order in which things happened to it — so the force in a pier under a sliding bearing is a record of its history, not a function of its loads.

Named alongside this one

Essays reaching for the same objects. Nobody chose these; they are what the concept index makes visible.

The objects this essay names

Each one links to every other essay that touches it.

DampingDuctility demandEnergy dissipationHysteresisImpulseNegative stiffnessNon-linear responseOverturningPrestressRestitutionRockingStability